Loop-space uniquely characterized by shifting of homotopy groups?

Question

Let $(X,x_{0})$ be a topological space and let $(Y,y_{0})$ be another topological space such that for homotopy groups

$\pi_{i}(Y,y_{0}) \cong \pi_{i+1}(X,x_{0})$

for all $i \geq 0$.

Is it then true that $Y \simeq \Omega X$ ?

I feel as if this is false in general, but you might be able to impose some conditions on $X$ and $Y$ to make this true.

(Or there might be some uniqueness of adjoints argument that makes this true in general).

Any help would be much appreciated!

Answer 1

The answer is no in general. A counter example is given by $S^3 \times \mathbb{C}P^\infty$ and $\Omega (S^2)$, by the Hopf fibration they have the shifted homotopy groups you ask for. However, $\Omega (S^3 \times \mathbb{C}P^\infty)=\Omega S^3 \times \Omega \mathbb{C}P^{\infty}$. One can use the Serre spectral sequence to see the cohomologies are not isomorphic.

There is not really a way to make this question have a positive answer, without begging the question. The context you need to answer questions about loop spaces is the notion of $E_n$ structures. The point is that any n-fold loop space has an $E_n$ structure and if two spaces are equivalent as spaces with $E_n$ structures with the additional property their $\pi_0$ forms a group, then these are both n-fold loop spaces of some third space (called the bar construction).

So you could add some of this information to your question and have it be true, but this is simply because for connected spaces looping and bar constructions are inverse. It might be the case that you could try to move the $E_n$ structure from the space to its $\Pi$-algebra and have an interesting question.

Answer 2

You may as well just set $Z = \Omega X$, where your question is now: "If $Y$ and $Z$ have the same homotopy groups, are they homotopy equivalent?"

You have surely seen counterexamples to this claim before. The standard one is $S^3 \times \Bbb{CP}^\infty$ and $S^2$. The first of these has the homotopy type of a loop space (it is $\Omega(\Bbb{HP}^\infty \times K(\Bbb Z, 3))$).

There are more or less only two good theorems of the kind you want.

1. Whitehead's theorem. If you have a continuous map which induces these isomorphisms, then that continuous map is a homotopy equivalence, provided both spaces are themselves homotopy equivalent to a CW complex.

2. If Y and Z each have a single nonzero homotopy group, and those groups are isomorphic, then $Y \simeq Z$. This is called uniqueness of Eilenberg MacLane spaces. It fails already for spaces with two nonzero homotopy groups. For instance, there are infinitely many inequivalent spaces $X_n$ with $\pi_2 \cong \pi_3 \cong \Bbb Z$ the only non-trivial homotopy groups.

They are distinguished by the Isomorphism class of $H^4$. These spaces have $H^4(X_n;\Bbb Z) \cong \Bbb Z/n$.